Single-axis pointing pure magnetic control algorithm for spacecraft based on geometrical analysis

ABSTRACT

Provided is a single-axis pointing pure magnetic control algorithm for a spacecraft based on geometrical analysis to realize single-axis pointing control of the spacecraft through the pure magnetic control algorithm in which a magnetic torque is only output by a magnetorquer to interact with a geomagnetic field to generate a control torque. The algorithm uses a spatial geometry method to obtain an optimally controlled magnetic torque direction, thereby designing a PD controller. The problem that the traditional magnetic control method is low in efficiency and even cannot be controlled is overcome. The algorithm is simple and easy, can be used in the attitude control field of spacecrafts, and achieves the pointing control in point-to-sun of a solar array and point-to-ground of antennae.

BACKGROUND OF THE INVENTION Technical Field

The invention relates to the technical field of spacecraft, inparticular to a single-axis pointing pure magnetic control algorithm fora spacecraft.

Description of Related Art

In a particular working mode, a spacecraft often needs to point an axison a body system to a specific location in space, such as pointing anegative normal axis of a solar array to the direction of a sun vector,and pointing a communication antenna to the center of the earth, etc.,which requires a single-axis pointing control. The problem to be solvedby the single-axis pointing control is as shown in FIG. 1. {right arrowover (O)}B in the figure is a unit vector in a certain axial direction(referred to as a body axis) under the body system of an aircraft; theunit vector {right arrow over (O)}A points to a certain target azimuthunder the inertial space (referred to as a target axis); the inclinedangle of the unit vector and the target azimuth is β; it is necessary togenerate a magnetic torque by a magnetorquer and then the magnetictorque interacts with a geomagnetic field to generate a control torque{right arrow over (T)} to control the rotation of the aircraft, andfinally the body axis coincides with the target axis. In the most idealcase (regardless of rotational inertia distribution and the currentangular velocity in each axial direction), the optimal direction of thedesired control torque {right arrow over (T)} is along the direction{right arrow over (OB)}×{right arrow over (OA)}, i.e., the direction{right arrow over (OD)} perpendicular to the plane AOB and satisfiedwith the right-hand rule (this direction is referred to as the optimalrotation axis). If a single-axis PD control rate is used in this case,the magnitude of {right arrow over (T)} can be calculated according tothe following equation:

T=K _(p) β−K _(d){dot over (β)}  (1)

where K_(p) and K_(d) are proportional differential controlcoefficients, respectively.

However, the truth is that the direction of the torque that themagnetorquer can output is limited by the direction of the geomagneticfield at the position where the aircraft is located, and cannot point toany direction, but only in the normal plane (the plane perpendicular toa geomagnetic field vector) of the local geomagnetic field. Due to theabove limitation, it is only possible to approximate T by a projectiontorque {right arrow over (T)}_(c) of the desired control torque T in thenormal plane of the geomagnetic field, where the desired torque T is acomponent of the actual control torque T_(c), as shown in FIG. 2. In thefigure, {right arrow over (T)}_(c) is the projection of {right arrowover (T)} in the normal plane of the geomagnetic field. Since {rightarrow over (T)} is not coincident with {right arrow over (T)}_(c),{right arrow over (T)}_(c) also generates an interference torquecomponent {right arrow over (T)}_(d) in addition to an effective controltorque component {right arrow over (T)}. In order to generate a magneticcontrol torque {right arrow over (T)}_(c), the magnetic torque {rightarrow over (M)} that the magnetorquer needs to output can be calculatedby equation (2):

$\begin{matrix}{\overset{->}{M} = \frac{\overset{->}{B} \times \overset{->}{T}}{{\overset{->}{B}}^{2}}} & (2)\end{matrix}$

where {right arrow over (B)} is the local geomagnetic field vector, theactual control torque {right arrow over (T)}_(c) generated in this caseis:

$\begin{matrix}{{\overset{->}{T}}_{c} = {{\overset{->}{M} \times \overset{->}{B}} = {\overset{->}{T} - {\left( {\frac{\overset{->}{B}}{B} \cdot \overset{->}{T}} \right)\frac{\overset{->}{B}}{B}}}}} & (3)\end{matrix}$

where T is the desired control torque, and

$\overset{->}{T} = {{- \left( {\frac{\overset{->}{B}}{B} \cdot \overset{->}{T}} \right)}{\frac{\overset{->}{B}}{B}.}}$

It can be seen from the above equation that the larger inclined angle ofthe magnetic field vector {right arrow over (B)} and the desired controltorque {right arrow over (T)} leads to a larger interference torque{right arrow over (T)}_(d), and the interference torque is zero when themagnetic field vector and the desired control torque are perpendicular.

In order to ensure the effectiveness of the control (the effect of theeffective control torque exceeds the effect of the interference torque),an effective control threshold is usually set, for example, the controloutput is generated only when T>T_(d) or the inclined angle of {rightarrow over (T)} and {right arrow over (T)}_(c) is less than 45°. In thefield of actual spacecraft engineering, limited by the direction of thegeomagnetic field, the above-mentioned threshold requirement oftencannot be reached in many positions on the spacecraft orbit. At thesepositions, the magnetic control algorithm will be in a stop state.

It can be seen from the above analysis that the existence of aninterference torque {right arrow over (T)}_(d) in the conventionalmagnetic control algorithm has a negative impact on the control effecton the one hand, and often causes a controller to be in a stop state onthe other hand. In addition, a rotation tendency (described by anangular acceleration) generated by the control amount is also related tothe rotational inertia of a rigid body, the angular velocity of therigid body, etc., which further increases the complexity of a controlsystem.

BRIEF SUMMARY OF THE INVENTION

The object of the invention is to provide a single-axis pointing puremagnetic control algorithm for a spacecraft based on geometric analysis,which solves the problem of high complexity of the control system in theprior art.

To achieve the above object, the invention is implemented by thefollowing technical solution:

A single-axis pointing pure magnetic control algorithm for a spacecraftbased on geometrical analysis: comprising the following steps:

{circle around (1)} acquiring the following data: coordinates of threevectors (a target azimuth vector {right arrow over (OA)}, a body axisvector {right arrow over (OB)} and a geomagnetic field vector {rightarrow over (B)}) in a body system and an aircraft inertia matrix I;

{circle around (2)} acquiring a pair of non-parallel vectors X and Y ina normal plane of a geomagnetic field;

{circle around (3)} calculating a unit normal vector Z′ of an angularacceleration plane:

X^(′) = I⁻¹X Y^(′) = I⁻¹Y$Z^{\prime} = \frac{X^{\prime} \times Y^{\prime}}{{X^{\prime} \times Y^{\prime}}}$

where I refers to the aircraft inertia matrix, X′ and Y′ are affinetransformed vectors of X and Y and are approximately in a plane of anangular acceleration generated by a magnetic torque;

{circle around (4)} acquiring an optimal control rotation axis {rightarrow over (OE)} by the following equation:

{right arrow over (OE)}=NORM(({right arrow over (OB)}+{right arrow over(OA)})×({right arrow over (OB)}×{right arrow over (OA)})×Z′)

where NORM is a vector normalization operator;

{circle around (5)} adjusting control coefficients and calculating amagnetic control torque T, wherein T must be in the normal plane of thegeomagnetic field in this case, and the magnetic control torque T iscalculated specifically by: calculating the magnitude and direction of arotation angular acceleration,

{dot over (ω)}_(r)=(K _(p) β{right arrow over (OE)})sign({right arrowover (OB)}·({right arrow over (O)}B×{right arrow over (O)}A))+K_(m)(ω·{right arrow over (OE)})

where K_(p) refers to a control coefficient for producing a rotationtendency in the direction {right arrow over (OE)}; β is the inclinedangle of the body axis {right arrow over (OB)} and the target axis{right arrow over (OA)}; sign is to obtain a positive/negative operatorwhich is valued as 1 when the dot product of the vectors {right arrowover (OE)} and {right arrow over (OD)} is positive or valued as −1 whenthe dot product is negative, or valued as 0 when the body axis and thetarget axis are perpendicular; the optimal rotation axis {right arrowover (OD)} is perpendicular to {right arrow over (OA)} and {right arrowover (OB)}; K_(m) refers to a control coefficient for limiting theangular velocity in the direction {right arrow over (OE)} fromovershoot; next, calculating a damped angular acceleration:

{dot over (ω)}_(d) =K _(d)[ω−(ω·Z′)Z′]

where K_(d) refers to a control coefficient for dampening an angularvelocity component in the angular acceleration plane; ω is theinstantaneous angular velocity of the star, (ω ·Z′)Z′ is a componentperpendicular to the angular acceleration plane in ω, so ω−(ω·Z′)Z′ is acomponent in the angular acceleration plane in ω; and then calculating atorque required to generate the rotation angular acceleration and thedamped angular acceleration:

T=I({dot over (ω)}_(r)+{dot over (ω)}_(d));

{circle around (6)} calculating the required control magnetic torque M:

$\overset{->}{M} = {\frac{\overset{->}{B} \times \overset{->}{T}}{{\overset{->}{B}}^{2}}.}$

The invention has a beneficial effect that the single-axis pointing puremagnetic control algorithm for a spacecraft based on geometricalanalysis of the invention realizes single-axis pointing control of thespacecraft through the pure magnetic control algorithm in which amagnetic torque is only output by a magnetorquer to interact with ageomagnetic field to generate a control torque. The algorithm overcomesthe problem that the traditional magnetic control method is low inefficiency and even cannot be controlled. The algorithm is simple andeasy, can be used in the control field of spacecrafts. The simulationresults further prove that the method can realize single-axis pointingcontrol such as point-to-ground of satellite antennas and thus can beused in a safety mode and specific control modes of spacecrafts.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

FIG. 1 is a schematic diagram of the single-axis pointing control ofFIG. 1.

FIG. 2 is a comparison diagram of a control torque generated by amagnetorquer and a desired control torque.

FIG. 3 is a schematic diagram of Lemma 1.

FIG. 4 is a schematic diagram of an optimal rotation axis direction.

FIG. 5 is a flow chart of the method of the invention.

FIG. 6 is a simulation result of a point-to-ground error angle curve.

FIG. 7 is a simulation result of an X-axis angular velocity curve.

FIG. 8 is a simulation result of a Y-axis angular velocity curve.

FIG. 9 is a simulation result of a Z-axis angular velocity curve.

DETAILED DESCRIPTION OF THE INVENTION

The invention will now be further described with reference to theaccompanying drawings and specific embodiments.

Lemma 1: Two unit vectors, {right arrow over (OA)} and {right arrow over(OB)}, are known. A unit vector {right arrow over (OC)} is on thebisecting line of the inclined angle formed by the above-mentioned twovectors. The unit vector

$\overset{}{OD} = \frac{\overset{}{OB} \times \overset{}{OA}}{{\overset{}{OB} \times \overset{}{OA}}}$

is perpendicular to the plane (Plane AOB) where {right arrow over (OA)}and {right arrow over (OB)} are located, {right arrow over (OB)} cancoincide with {right arrow over (OA)} after rotating a certain anglearound any rotation axis passing the point O and located in Plane DOC.

It is proved as follows: Any unit vector axis {right arrow over (OE)}passing O and located in the plane DOC is selected, then there must beconstants a and b, and then:

{right arrow over (OE)}=a{right arrow over (OC)}+b{right arrow over(OD)}  (4)

Because the unit vector {right arrow over (OC)} is on the bisecting lineof ∠AOB, and then

$\begin{matrix}{\overset{}{OC} = \frac{\overset{}{OA} + \overset{}{OB}}{{\overset{}{OA} + \overset{}{OB}}}} & (5)\end{matrix}$

Let ∥{right arrow over (OA)}+{right arrow over (OB)}∥=c, (5) issubstituted into (4), and then,

$\begin{matrix}{\overset{}{OE} = {{\frac{a}{c}\left( {\overset{}{OA} + \overset{}{OB}} \right)} + {b\overset{}{OD}}}} & (6)\end{matrix}$

The dot products of the vectors are further calculated as follow:

$\begin{matrix}{{\overset{}{OE} \cdot \overset{}{OA}} = {{\frac{a}{c}\left( {{\overset{}{OA} \cdot \overset{}{OA}} + {\overset{}{OB} \cdot \overset{}{OA}}} \right)} + {b{\overset{}{OD} \cdot \overset{}{OA}}}}} & (7) \\{{\overset{}{OE} \cdot \overset{}{OB}} = {{\frac{a}{c}\left( {{\overset{}{OA} \cdot \overset{}{OB}} + {\overset{}{OB} \cdot \overset{}{OB}}} \right)} + {b{\overset{}{OD} \cdot \overset{}{OB}}}}} & (8)\end{matrix}$

Since each vector is a unit vector and {right arrow over (OD)} isperpendicular to {right arrow over (OA)} and {right arrow over (OB)},the following results are established:

{right arrow over (OD)}·{right arrow over (OA)}={right arrow over(OD)}·{right arrow over (OB)}=0

{right arrow over (OA)}·{right arrow over (OA)}={right arrow over(OB)}·{right arrow over (OB)}=1

{right arrow over (OA)}·{right arrow over (OB)}={right arrow over(OB)}·{right arrow over (OA)}  (9)

(9) is substituted into (7) and (8), and then,

{right arrow over (OE)}·{right arrow over (OA)}={right arrow over(OE)}·{right arrow over (OB)}  (10)

Thus it is concluded that ∠AOE=∠BOE, that is to say, {right arrow over(OB)} must coincide with {right arrow over (OA)} after rotating acertain angle around {right arrow over (OE)}, and the lemma is proved.

The main function of the lemma is to expand the selection range of therotation axis.

In addition to the optimal rotation axis {right arrow over (OD)},rotation around a series of sub-optimal rotation axes in the same planealso can result in that the body axis points to the target axis. For theconvenience, the plane DOC is referred to as a rotation axis plane. Itshould be noted that when the inclined angle of {right arrow over (OE)}and {right arrow over (OD)} is less than 90°, the angle of rotationrequired is less than 180°; when the inclined angle is larger than 90°,the angle of rotation required is larger than 180°, and the angle ofrotation required is exactly 180° when they are perpendicular.

According to the rigid body dynamics equation, it is assumed that therotational inertia matrix of the aircraft is I and the instantaneousangular velocity is ω, the angular acceleration of the aircraft underthe control torque T is calculated as follows:

{dot over (ω)}=I ^(−i)(T−ω×Iω)  (11)

When the instantaneous angular velocity of the star is small, the itemsin ω×Iω are all second-order infinitesimal and can be approximatelyignored, and then

{dot over (ω)}≈I ⁻¹ T  (12)

Since the torque T is located in the normal plane of the localgeomagnetic field B, T must be linearly represented by a pair ofnon-parallel vectors X and Y in the normal plane, that is, for anymagnetic control torque T, the following is must concluded:

T=aX+bY  (13)

where a and b are constant coefficients; the constants are substitutedinto (12), then

{dot over (ω)}≈a(I ⁻¹ X)+b(I ⁻¹ Y)=aX′+bY′  (14)

According to the parallel retention of affine coordinate transformation,since X and Y are not parallel, X′ and Y′ must also be non-parallelafter affine transformation, an angular acceleration generated by anytorque T in the normal plane (the plane formed by X and Y) of themagnetic field is located in the plane formed by the vectors X′ and Y′(the plane is simply referred to as the angular acceleration plane),thereby forming one-to-one mapping from the normal plane (also regardedas a magnetic torque plane) of the magnetic field to the angularacceleration plane by affine coordinate transformation. A unit normalvector of the angular acceleration plane may be obtained according tothe following equation:

$\begin{matrix}{Z^{\prime} = \frac{X^{\prime} \times Y^{\prime}}{{X^{\prime} \times Y^{\prime}}}} & (15)\end{matrix}$

Obviously, Z′ is an ideal normal vector of the angular accelerationplane.

In summary, when the instantaneous angular velocity is very small, theangular acceleration generated by the magnetic control torque in thenormal plane of the magnetic field is approximately located in theangular acceleration plane. According to the conclusion of Lemma 1, theangular acceleration control can be performed according to the followingmethod:

(1) when the angular acceleration plane is parallel to the rotation axisplane, a rotation angular acceleration {dot over (ω)}, can be generatedalong the direction of the optimal rotation axis {right arrow over(OD)}, so that a tendency of rotating {right arrow over (OA)} to {rightarrow over (OB)} is generated;

(2) when the angular acceleration plane is not parallel to the rotationaxis plane, the two planes must have an intersection line, and arotation angular acceleration {dot over (ω)}_(r) is generated along theintersection line {right arrow over (OE)}, and a tendency of rotating{right arrow over (OA)} to {right arrow over (OB)} is also generated, asshown in FIG. 4;

(3) the modulus value of the rotation angular acceleration {dot over(ω)}_(r) should be proportional to the inclined angle of {right arrowover (OA)} and {right arrow over (OB)} and moreover the problem thatlarge overshoot is caused by the angular velocity on {right arrow over(OE)} can be prevented;

(4) in order to ensure the system stability, angular velocity dampingshould be performed, that is, a damped angular acceleration {dot over(ω)}_(d) is generated in the opposite direction of the projection of theinstantaneous angular velocity of the aircraft in the angularacceleration plane, and the modulus value of the damped angularacceleration is proportional to the modulus value of the correspondingprojection;

(5) the angular acceleration generated by the actual control torqueoutput should be the vector sum of the rotation angular acceleration{dot over (ω)}_(r) and the damped angular acceleration {dot over(ω)}_(d).

The direction of rotation angular acceleration {dot over (ω)}_(r) can becalculated according to the following equation:

$\begin{matrix}\begin{matrix}{\overset{}{OE} = {{NORM}\left( {\overset{}{OC} \times \overset{}{OD} \times Z^{\prime}} \right)}} \\{= {{NORM}\left( {\left( {\overset{}{OB} + \overset{}{OA}} \right) \times \left( {\overset{}{OB} \times \overset{}{OA}} \right) \times Z^{\prime}} \right)}}\end{matrix} & (16)\end{matrix}$

where NORM is a vector normalization operator. Since the vector {rightarrow over (OC)}×{right arrow over (OD)} is the normal of the rotationaxis plane and Z′ is the normal of the angular acceleration plane, so{right arrow over (OE)} is perpendicular to the normals of both therotation axis plane and the angular acceleration plane, that is, both inthe rotation axis plane and the angular acceleration plane (the twoplanes intersect with each other), X and Y are any two non-parallelvectors in the normal plane of the magnetic field. The magnitude anddirection of the rotation angular acceleration can be calculated asfollows:

Ω_(r)=(K _(p) β{right arrow over (OE)})sign({right arrow over(OE)}·({right arrow over (OB)}×{right arrow over (OA)}))+K _(m)(ω·{rightarrow over (OE)})  (17)

where K_(p) refers to a control coefficient; β is the inclined angle ofthe body axis and the target axis; sign is to obtain a positive/negativeoperator which is valued as 1 when the dot product of the vectors {rightarrow over (OE)} and {right arrow over (OD)} is positive (the inclinedangle is less than 90°) or valued as −1 when the dot product isnegative, or valued as 0 when the body axis and the target axis areperpendicular; K_(m) refers to a control coefficient for controlling theangular velocity in the direction {right arrow over (OE)}. A dampedangular acceleration can be calculated according to the followingequation:

{dot over (ω)}_(d) =K _(d)[ω−(ω·Z′)Z′]  (18)

where K_(d) refers to a control coefficient for dampening an angularvelocity component in the angular acceleration plane; ω is theinstantaneous angular velocity of the star, (ω·Z′)Z′ is a componentperpendicular to the angular acceleration plane in ω, so ω−(ω·Z′)Z′ is acomponent in the angular acceleration plane in ω. A torque required togenerate the rotation angular acceleration and the damped angularacceleration is then calculated as follows:

T=I(ωr+ωd)  (19)

where I is the rotational inertia of the spacecraft. It can be seen fromthe above derivation that since {dot over (ω)}_(r)+{dot over (ω)}_(d) isin the angular acceleration plane, the torque T must be in the normalplane of the magnetic field. Then, the required control magnetic torqueM can be obtained by the equation (2). Since the torque T isperpendicular to the magnetic field B, it can be completely generated bya magnetorquer.

In summary, as shown in FIG. 5, the implementation steps of thesingle-axis pointing magnetic control algorithm designed by theinvention are as follows:

{circle around (1)} acquiring the following data: coordinates of threevectors, a target azimuth vector {right arrow over (OA)}, a body axisvector {right arrow over (OB)} and a geomagnetic field vector {rightarrow over (B)}, in a body system and an aircraft inertia matrix I;

{circle around (2)} acquiring a pair of non-parallel vectors X and Y ina normal plane of a geomagnetic field, and calculating a cross productof {right arrow over (B)} and any vector that is not parallel to thegeomagnetic field vector {right arrow over (B)} to obtain a vector X,and calculating a cross product of X and {right arrow over (B)} toobtain Y;

{circle around (3)} acquiring a unit normal vector Z′ of an angularacceleration plane according to the equations (14) and (15);

{circle around (4)} acquiring {right arrow over (OE)} according to theequation (16);

{circle around (5)} adjusting control coefficients and calculating amagnetic control torque T according to the equations (17), (18) and(19), wherein T must be in the normal plane of the magnetic field inthis case; and

{circle around (6)} calculating the required control magnetic torque Maccording to the equation (2).

The algorithm of the invention is essentially a PD control method basedon geometric analysis and affine coordinate transformation. Thealgorithm is simple and easy to implement, and can be completely appliedto practical engineering applications.

The simulation test is carried out under Matlab. The pure magneticcontrol is used to cause the Z axis of the star to point to the centerof the earth to form a slowly changing follow-up system. The simulationparameters are shown in Table 1:

TABLE 1 Simulation parameters Orbital attitude 500 Km Orbitalinclination 30° Orbital eccentricity 0 Initial pointing deviation 92°Each axial initial angular velocity  0.2°/s Three-axis residualmagnetism 0.6 Am² Maximum output of magnetorquer 25 Am² Kp 0.001 Km−0.001 Kd −0.01

The simulation results are as shown in FIG. 6, FIG. 7, FIG. 8 and FIG.9. When the pointing error longitude is converged to be less than 1° andthe angular velocity of each axis is also controlled when the attitudeis less than 10000 s. The simulation is carried out under the conditionthat other simulation parameters are unchanged and only the orbitalinclination is modified. It is confirmed that the method of theinvention can obtain good control results under the orbital inclinationranging from 30° to 90°, only with the difference in convergence time.

The foregoing is a further detailed description of the invention inconnection with the specific preferred embodiments, and it is not to bedetermined that the specific embodiments of the invention are limited tothese descriptions. Multiple simple deductions or replacements made bythose skilled in the field to which the invention belongs, withoutdeparting from the spirit and scope of the invention, shall beconsidered as falling within the scope of the invention.

1. A single-axis pointing pure magnetic control algorithm for aspacecraft based on geometrical analysis, comprising the followingsteps: {circle around (1)} acquiring following data: coordinates of atarget azimuth vector {right arrow over (OA)}, a body axis vector {rightarrow over (OB)} and a geomagnetic field vector {right arrow over (B)}in a body system and an aircraft inertia matrix I; {circle around (2)}acquiring a pair of non-parallel vectors X and Y in a normal plane of ageomagnetic field; {circle around (3)} calculating a unit normal vectorZ′ of an angular acceleration plane through affine transformation of theaircraft inertia matrix I: X^(′) = I⁻¹X Y^(′) = I⁻¹Y$Z^{\prime} = \frac{X^{\prime} \times Y^{\prime}}{{X^{\prime} \times Y^{\prime}}}$wherein X′ and Y′ are affine transformed vectors of X and Y and areapproximately in a plane of an angular acceleration generated by amagnetic torque; {circle around (4)} acquiring an optimal controlrotation axis {right arrow over (OE)} by a following equation:{right arrow over (OE)}=NORM(({right arrow over (OB)}+{right arrow over(OA)})×({right arrow over (OB)}×{right arrow over (OA)})×Z′) where NORMis a vector normalization operator; {circle around (5)} adjustingcontrol coefficients and calculating a magnetic control torque T,wherein T must be in the normal plane of the magnetic field in thiscase, and the magnetic control torque T is calculated specifically by:calculating a magnitude and a direction of a rotation angularacceleration,ω_(Y)=(K _(p) β{right arrow over (OE)})sign({right arrow over(OE)}·({right arrow over (OB)}×{right arrow over (OA)}))+K _(m)(ω·{rightarrow over (OE)}) wherein K_(p) refers to a control coefficient; β is aninclined angle between a body axis {right arrow over (OB)} and a targetazimuth {right arrow over (OA)}; sign is to obtain a positive/negativeoperator which is valued as 1 when a dot product of the vectors {rightarrow over (OE)} and {right arrow over (OD)} is positive or valued as −1when the dot product is negative, or valued as 0 when the body axis andthe target axis are perpendicular; an optimal rotation axis {right arrowover (OD)} is perpendicular to {right arrow over (OA)} and {right arrowover (OB)}; K_(m) refers to a control coefficient for controlling anangular velocity in the direction {right arrow over (OE)}; next,calculating a damped angular acceleration:{dot over (ω)}_(d) =K _(d)[ω−(ω·Z′)Z′] wherein K_(d) refers to a controlcoefficient; ω is the instantaneous angular velocity of the star,(ω·Z′)Z′ is a component perpendicular to the angular acceleration planein ω, so ω−(ω·Z′)Z′ is a component in the angular acceleration plane inω; and then calculating a torque required to generate the rotationangular acceleration and the damped angular acceleration:T=I({dot over (ω)}_(r)+{dot over (ω)}_(d)); and {circle around (6)}calculating a required control magnetic torque M:$\overset{->}{M} = {\frac{\overset{->}{B} \times \overset{->}{T}}{{\overset{->}{B}}^{2}}.}$2. The single-axis pointing pure magnetic control algorithm according toclaim 1, wherein the step {circle around (2)} is specifically:calculating a cross product of {right arrow over (B)} and any vectorthat is not parallel to the geomagnetic field vector {right arrow over(B)} to obtain a vector X, and calculating a cross product of X and{right arrow over (B)} to obtain Y.
 3. An attitude control method for aspacecraft, wherein the method employs the single-axis pointing puremagnetic control algorithm according to claim
 1. 4. An attitude controlmethod for a spacecraft, wherein the method employs the single-axispointing pure magnetic control algorithm according to claim 2.